Fundamental Groups Of Positively Curved Manifolds

This thesis is based on parts of the report in the seminar of "positive curvature, symmetry and topology" in the fall semester of 2006 at Capital Normal University. We first briefly review basic notions and theorems related to this thesis. Then we will collect some results on "Fundamental group of positively curved manifolds" and we will investigate some problems arose.The main results in this thesis are: Theorems A (the inverse problem of the Myers's theorem): "Any finite group can be the fundamental group of some closed manifold of positive Ricci curvature." Professor Xiaochun Rong has sketched a proof in his unpublished lecture notes. Here we use the method of group's extension to give a detailed proof; Theorems B: "Let M be a closed manifold of positive sectional curvature on which a torus T~k(k≥1) acts isometrically. If φ is an isometry on M commuting with the T~k-action, then φ preserves some T~k-orbit which is a circle." This theorem is a basic result on "isometric T~k-action on positively curved manifolds", which establish as a bridge between an isometric torus group action on a manifold with positive sectional curvature and its fundamental group. Professor Rong gave a proof for odd dimension case in [12] and asserted that the theorem still holds in even dimensions. (though even-dimensional case may not be needed as long as fundamental groups are concerned) In this thesis, we will give a unify proof of the theorem in both odd and even dimensional cases. Theorem C, D and E are important applications of Theorems B in [11]and [12]. For the completeness, we will also give self-contained proof, in details.